Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

Abstract

We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose space x and time t variables are a function of an evolution parameter τ . The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter τ . We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by τ ) is generalized onto a 1 , 2 -dimensional supermanifold which is characterized by the superspace coordinates Z M = τ , θ , θ ¯ where a pair of the Grassmannian variables satisfy the fermionic relationships: θ 2 = θ ¯ 2 = 0 , θ θ ¯ + θ ¯ θ = 0 , and τ is the bosonic evolution parameter. In the context of ACSA, we take into account only the 1 , 1 -dimensional (anti)chiral super submanifolds of the general 1 , 2 -dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.

BRST and Related Superfield Approach to 1D and 2D Diffeomorphism Invariant Models of Particles and a Bosonic String: A Brief Review

“…The CF-type restriction(s) in the 1D and 2D diffeomorphism invariant models are the special cases of the CF-type restrictions for the D-dimensional diffeomor-phism invariant theory. Finally, we have not discussed (in our previous works [21][22][23]), the 1D diffeomorphism invariant interacting scalar relativistic particle where the interacting electromagnetic field is constrained to remain in the background. It is a challenging problem for us to show that (i) the form of the (anti-)BRST invariant CF-type restriction, and (ii) the form of the gauge-fixing and FP ghost terms remain the same as in the cases of the 1D diffeomorphism invariant free (non-)SUSY and (non-)relativistic particles [21][22][23].…”

“…[28,21]), we have found three equivalent reparameterization-invariant Lagrangians for this generalized version of a free non-relativistic particle as (see, e.g. [21,27,28] for details)…”

“…First of all, we have recently written a set of papers (see, e.g. [21][22][23][24][25]) where we have discussed the 1D and 2D diffeomorphism symmetries at the classical level and their counterparts at the quantum level within the framework of BRST formalism. Thus, it is interesting to collate all these results together and present them in a unified and coherent fashion.…”

“…The full set of off-shell nilpotent [s 2 (a)b = 0] and absolutely anticommuting (s b s ab + s ab s b = 0) (anti-)BRST symmetry transformations [s (a)b ] for our system are as follows (see, e.g. [21,28] for details):…”

“…The full set of (anti-)BRST symmetry transformations [s (a)b ], corresponding to the above classical reparameterization symmetry transformations (9), are (see, e.g. [21] for details)…”